1. Field of the Invention
The present invention relates to a lithographic apparatus, a method for controlling a parameter of the lithographic apparatus, a control system and a method for dimensioning a transfer function in such a control system or lithographic apparatus.
2. Description of the Related Art
Lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). In such a case, a patterning device may be used to generate a desired circuit pattern corresponding to an individual layer of the IC, and this pattern can be imaged onto a target portion (e.g. comprising one or more dies) on a substrate (silicon wafer) that has been coated with a layer of radiation-sensitive material (resist).
In general, a single substrate will contain a network of adjacent target portions that are successively exposed. Known lithographic apparatus include so-called steppers, in which each target portion is irradiated by exposing an entire pattern onto the target portion in one go, and so-called scanners, in which each target portion is irradiated by scanning the pattern through the projection beam in a given direction (the “scanning”—direction) while synchronously scanning the substrate parallel or anti-parallel to this direction.
Conventional control systems that control a position of a moveable part of an apparatus, often employ a control configuration that comprises a feed-forward and a feed-back controller. The feed-back, in a form of a feed-back loop, accounts for a high accuracy, a low steady state error and robustness against, for example, product variations and disturbances. The feed-forward accounts for a high speed of the control system and a fast response. Such control systems can be implemented in an analog form, such as, for example, analog electronics.
Alternatively, a part or all of the control system can be implemented in a numeric, i.e., a digital form. The numeric or digital parts can be implemented on a computer, controller or any other suitable numeric device. In an analog implementation a transfer function of the control system (or of parts thereof) can be expressed in the Laplace or s-domain, while in a numeric implementation such transfer function can be expressed in the z-domain.
The control system can be used for controlling any physical quantity, thus not only being limited to a control system for controlling a position of a movable part, however, can also be applied for e.g. a velocity, acceleration, a temperature, a slow, a light intensity, elimination, or any other physical quantity. In a lithographic apparatus, requirements on accuracy as well as on speed of the control system are generally strict. For this reason, a high performance control configuration comprising a feed-forward and a feed-back is frequently applied in a lithographic apparatus.
To obtain an optimal feed-forward, a transfer function of the feed-forward should comprise an inverse of a transfer function of the actuator, which is controlled by the control system. Hence, in the s-domain, if the actuator comprises an integrative transfer function, the feed-forward should comprise a differentiator, while in the z-domain if the actuator comprises a delay, the feed-forward should comprise a time lead thus compensating for the delay.
A problem that arises is that the inverse of the transfer function of the actuator is not always stable. In the s-domain, a pole in the transfer function of the actuator leads to a zero in the transfer function of the feed-forward and vice versa. Thus, if in the s-domain the transfer function of the actuator has a zero in the left half plane, this lead to a pole in the left half plane in the transfer function of the feed-forward. Thus, the transfer function of the feed-forward in the s-domain would be unstable. In a numeric implementation, in the z-domain, if the transfer function of the actuator has a zero which falls outside the unity circle in the z-domain, this will result in a pole outside the unity circle in the transfer function of the feed-forward. Thus, an unstable function of the feed-forward would result.
A known solution to avoid obtaining an unstable feed-forward transfer function in the numeric case, is to mirror the pole in the transfer function of the feed-forward which lies outside the unity circle, into the unity circle. Thus, if a pole in the feed-forward transfer function occurs as z=−4 than it is mirrored to z=−¼, thus being mirrored from outside the unity circle into the unity circle hence resulting in a stable transfer function in the feed-forward. A disadvantage of this solution, however, is that by using this mirroring operation, an inaccurate approximation of the feed-forward is created, since the operation, especially the phase performance of the feed-forward, has been changed. Thus, by this mirroring operation, a stable transfer function of the feed-forward is created but at the cost of a decrease in performance of the control system.
Whether or not a transfer function in the z-domain is unstable does not solely depend on the transfer function having a pole outside the unity circle. For example, ratios between a delay time of the actuator (and hence the position of the pole response zero) and a sample time of the numeric system (i.e., the time corresponding to a unity delay) may influence pole instability. Hence, in this application, when reference is made to an unstable pole, this is to be interpreted as also comprising a potentially unstable pole, thus a pole which could become unstable depending on the ratio between the position of the pole and the sample time.